In other words, if we hold the first entry of the bilinear map fixed, while letting the second entry vary, the result is a linear operator, and similarly if we hold the second entry fixed. Note that if we regard the product V × W as a vector space, then B is not a linear transformation of vector spaces (unless V = 0 or W = 0) because, for example B(2(v,w)) = B(2v,2w) = 2B(v,2w) = 4B(v,w).

If V = W and we have B(v,w) = B(w,v) for all v, w in V, then we say that B is symmetric.

The definition works without any changes if instead of vector spaces over a field F, we use modules over a commutative ringR. It also can be easily generalized to n-ary functions, where the proper term is multilinear.

For the case of a non-commutative base ring R and a right module MR and a left module RN, we can define a bilinear map B : M × N → T, where T is an abelian group, such that for any n in N, m ↦ B(m, n) is a group homomorphism, and for any m in M, n ↦ B(m, n) is a group homomorphism too, and which also satisfies

A first immediate consequence of the definition is that B(x,y) = 0 whenever x = 0 or y = 0. (This is seen by writing the null vector0 as 0·0 and moving the scalar 0 "outside", in front of B, by linearity.)

A matrix M determines a bilinear map into the real by means of a real bilinear form (v,w) ↦ v′Mw, then associates of this are taken to the other three possibilities using duality and the musical isomorphism

If V, W, X are finite-dimensional, then so is L(V,W;X). For X = F, i.e. bilinear forms, the dimension of this space is dim V × dim W (while the space L(V×W;F) of linear forms is of dimension dim V + dim W). To see this, choose a basis for V and W; then each bilinear map can be uniquely represented by the matrix B(ei,fj), and vice versa. Now, if X is a space of higher dimension, we obviously have dim L(V,W;X) = dim V × dim W × dim X.

The null map, defined by B(v,w) = 0 for all (v,w) in V × W is the only map from V × W to X which is bilinear and linear at the same time. Indeed, if (v,w) ∈ V × W, then if B is linear, B(v,w) = B(v,0) + B(0,w) = 0 + 0 if B is bilinear.